Algebra and fractions

Hi all, I came across this problem and its solution:

There are 5 people A-E. They have alot of oranges. First, A takes 1 orange from the basket, and 1/5 of the remaining. Next, B takes 1 orange, and 1/5 of the remainder again. And So on. What is the least number of oranges that they have?

Basically the solution is 5^5-4.

I came up with a general case whereby

There are n people. They have alot of oranges. First, A takes 1 orange from the basket, and 1/n of the remaining. Next, B takes 1 orange, and 1/n of the remainder again. And So on. What is the least number of oranges that they have?

And the solution should be n^n - (n-1)

I understand the n^n but as for the n-1, I couldn't figure out the logic in it. Could anyone help me with this?